Question

If \([A]_{3 \times 2} [B]_{x \times y} = [C]_{3 \times 1}\), then \( x \) and \( y \) are:

• A. $x = 1, y = 3$
• B. $x = 2, y = 1$
• C. $x = 3, y = 3$
• D. $x = 3, y = 1$

Hint:

When performing matrix multiplication, it's important to ensure that the number of columns in the first matrix matches the number of rows in the second matrix. Additionally, the dimensions of the resulting product matrix are determined by the number of rows from the first matrix and the number of columns from the second matrix. Make sure to check these conditions before proceeding with the multiplication.

Answer 1

The Correct Option is B

Given the matrices:

\[ [A]_{3 \times 2}, \quad [B]_{x \times y}, \quad [C]_{3 \times 1}. \]

For matrix multiplication \([A][B]\) to be defined, the number of columns of \(A\) must equal the number of rows of \(B\), so:

\[ x = 2. \]

The resulting product \([A][B]\) will have dimensions \(3 \times y\), which must match \([C]_{3 \times 1}\), so:

\[ y = 1. \]

Thus:

\[ x = 2, \quad y = 1. \]

The correct option is:

\[ x = 2, \, y = 1. \]

Answer 2

Given the matrices: 

\[ [A]_{3 \times 2}, \quad [B]_{x \times y}, \quad [C]_{3 \times 1}. \]

Step 1: Check the condition for matrix multiplication \([A][B]\) to be defined:

For matrix multiplication \([A][B]\) to be defined, the number of columns of \(A\) must equal the number of rows of \(B\). Since matrix \(A\) is \(3 \times 2\), the number of columns is 2. Therefore, the number of rows of matrix \(B\) must be:

\[ x = 2. \]

Step 2: Consider the dimensions of the resulting product \([A][B]\):

The resulting product \([A][B]\) will have dimensions determined by the number of rows of \(A\) and the number of columns of \(B\). Matrix \(A\) has 3 rows and matrix \(B\) has \(y\) columns. Therefore, the resulting product \([A][B]\) will have dimensions \(3 \times y\). This product must match the dimensions of matrix \(C\), which is \(3 \times 1\), so:

\[ y = 1. \]

Step 3: Conclusion:

Thus, the values of \(x\) and \(y\) are:

\[ x = 2, \quad y = 1. \]

The correct option is:

\[ x = 2, \, y = 1. \]